**Student Mathematical Library**

Volume: 42;
2008;
262 pp;
Softcover

MSC: Primary 37; 28; 54;

Print ISBN: 978-0-8218-4420-5

Product Code: STML/42

List Price: $49.00

Individual Price: $39.20

**Electronic ISBN: 978-1-4704-2151-9
Product Code: STML/42.E**

List Price: $49.00

Individual Price: $39.20

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#### Supplemental Materials

# Invitation to Ergodic Theory

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*C. E. Silva*

This book is an introduction to basic concepts in ergodic theory such as recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It does not assume knowledge of measure theory; all the results needed from measure theory are presented from scratch. In particular, the book includes a detailed construction of the Lebesgue measure on the real line and an introduction to measure spaces up to the Carathéodory extension theorem. It also develops the Lebesgue theory of integration, including the dominated convergence theorem and an introduction to the Lebesgue \(L^p\)spaces.

Several examples of a dynamical system are developed in detail to illustrate various dynamical concepts. These include in particular the baker's transformation, irrational rotations, the dyadic odometer, the Hajian–Kakutani transformation, the Gauss transformation, and the Chacón transformation. There is a detailed discussion of cutting and stacking transformations in ergodic theory. The book includes several exercises and some open questions to give the flavor of current research. The book also introduces some notions from topological dynamics, such as minimality, transitivity and symbolic spaces; and develops some metric topology, including the Baire category theorem.

#### Table of Contents

# Table of Contents

## Invitation to Ergodic Theory

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface vii8 free
- Chapter 1. Introduction 111 free
- Chapter 2. Lebesgue Measure 515 free
- §2.1. Lebesgue Outer Measure 515
- §2.2. The Cantor Set and Null Sets 1020
- §2.3. Lebesgue Measurable Sets 1727
- §2.4. Countable Additivity 2333
- §2.5. Sigma-Algebras and Measure Spaces 2636
- §2.6. The Borel Sigma-Algebra 3444
- §2.7. Approximation with Semi-rings 3848
- §2.8. Measures from Outer Measures 4757

- Chapter 3. Recurrence and Ergodicity 5969
- §3.1. An Example: The Baker's Transformation 6070
- §3.2. Rotation Transformations 6777
- §3.3. The Doubling Map: A Bernoulli Noninvertible Transformation 7585
- §3.4. Measure-Preserving Transformations 8393
- §3.5. Recurrence 8696
- §3.6. Almost Everywhere and Invariant Sets 91101
- §3.7. Ergodic Transformations 95105
- §3.8. The Dyadic Odometer 102112
- §3.9. Infinite Measure-Preserving Transformations 109119
- §3.10. Factors and Isomorphism 115125
- §3.11. The Induced Transformation 120130
- §3.12. Symbolic Spaces 123133
- §3.13. Symbolic Systems 127137

- Chapter 4. The Lebesgue Integral 131141
- §4.1. The Riemann Integral 131141
- §4.2. Measurable Functions 134144
- §4.3. The Lebesgue Integral of Simple Functions 141151
- §4.4. The Lebesgue Integral of Nonnegative Functions 145155
- §4.5. Application: The Gauss Transformation 150160
- §4.6. Lebesgue Integrable Functions 155165
- §4.7. The Lebesgue Spaces: L[sup(1)],L[sup(2)] and L∞ 159169
- §4.8. Eigenvalues 166176
- §4.9. Product Measure 170180

- Chapter 5. The Ergodic Theorem 175185
- Chapter 6. Mixing Notions 201211
- Appendix A. Set Notation and the Completeness of R 235245
- Appendix B. Topology of R and Metric Spaces 241251
- Bibliographical Notes 251261
- Bibliography 255265
- Index 259269
- Back Cover Backcover1273

#### Readership

Undergraduate and graduate students interested in ergodic theory and measure theory.

#### Reviews

...comprehensive in scope, uncovering key ideas ranging from Euclidean geometry to transformations to affine systems to non-Euclidean geometries. ...The authors neither cut corners nor 'wave' at neat ideas; rather, they try to connect everything via a quite rigorous development, complete with well-chosen exercises.

-- CHOICE Reviews

The writing is crisp and clear. Proofs are written carefully with adequate levels of detail. Exercises are plentiful and well-integrated with the text.

-- MAA Reviews

I can only warmly recommend this book to students or as the basis for a course.

-- Monatshafte für Mathematik

The author presents in a very pleasant and readable way an introduction to ergodic theory for measure-preserving transformations of probability spaces. In my opinion, the book provides guidelines, classical examples and useful ideas for an introductory course in ergodic theory to students that have not necessarily already been taught Lebesgue measure theory.

-- Elemente der Mathematik

The book contains many (often easy or very easy) exercises, both in the text as well as at the end of each section.

-- Mathematical Reviews